Math-MathEd 300 Homework, Fall Semester, 2009

1. Aug 31. Make a table and express the numbers 574 and 475 in terms of the following numeral systems: Egyptian,

Roman, Babylonian, Chinese-Japanese, Greek, Mayan, and multiplicative grouping, base 5. Due Sep 4.

2. Sep 2. 1. Do the following operations, where the numbers are given base 5: 4213 + 1334, 4214 * 324.

2. Express the integer 36972, given in base 10, in base 7.

3. The number 251634 is in base 7. Express this number in base 5. Due Sep 9

3. Sep 4. This exercise is to find Pythagorean triples.

1. Find all triples all of whose terms are less than or equal to 50

a. By the Pythagorean formulas

b. By the Plato formulas

2. Find 25 Colin Gardner triples. Due Sep 11

4. Sep 9. 1. Verfy that 1184 and 1210 form an amicable pair

2. Show that 945 is an abundant number

3. Show how the following can be established geometrically:

a. a(b + c) = ab + ac

b. (a + b)(c + d) = ac + bc + ad + bd

c. a^2 – b^2 = (a + b)(a – b) Due Sep 14

5. Sep 11. 1. Plutarch, the Greek historian, noted that eight times a triangular number plus one unit is a

square number; this fact would later be used by Diophantus. Verify this statement.

2. Verify that:

a. A square number is either divisible by three, or will be divisible by three after the

subtraction of a unit.

b. A square number is either divisible by four, or will be divisible after the subtraction

of a unit.

c. The sum of the first n odd numbers is the nth square number

3. Verify that (3,4,5) is the only Pythagorean triple involving consecutive positive integers.

(Hint: what would happen if (x, x + 1, x + 2) were such a triple?)

4. Show that there are infinitely many Pythagorean triples (x,y,z) in which x and y are

consecutive integers. (Hint: verify that if (x, x + 1, z) is a triple, so is (3x + 2z + 1, 3x + 2z + 2, 4x + 3z + 2).) Due Sep 16

6. Sep 14. 1. Let p be any prime number. Prove that the square root of p is irrational.

2. Let log(x) represent the logarithm base 10 of x. Show that log(2) is irrational

3. The Euclidean Algorithm is a process for finding the greatest common divisor (gcd) of two numbers.

It uses the division algorithm; where a = qb + r; a divided by b has quotient q and remainder r, 0<r<b.

a. Explain how to use the division algorithm repeatedly to obtain the gcd of two numbers a and b.

b. Find the gcd of 210 and 5148 by your method in part a. Due Sep 18

7. Sep 16. The Archimedes-Brahmagupta (A-B) equation, later called Pell’s equation, is A’s cattle problem leads to this

equation, with a very large value of m. B discovered that if (x,y) and (x’,y’) are two solutions to this equation, that so is

(xx’ + myy’,xy’ + x’y) a solution.

Problem 1. A. Prove the above statement.

B. So if we have one non-trivial solution (note that (1,0) is the trivial solution), verify that we can get infinitely many solutions.

Problem 2. For the equation , show that (3,2) is a solution and find three more solutions.

Problem 3. Find three non-trivial solutions to the equation Due Sep 21

8. Sep 18. Sketch 1, Neil McLeod, Scott Sprague. Problems 1 and 2, page 71. Due Sep 23.

9. Sep 21. Sketch 2, Marie Beck. Page 77: Problems 1, 2, 3a. Due Sep 25.

10. Sep 23, Sketch 3, Stephanie Carruth. Page 83; 1 and 4, due Sep 28.

11. Sep 25, Class homework, due Friday, Oct 2. 1. A). Create a magic square of order 7. B). Create a magic square of order 3, with a magic number of 21. 2. Check the following sum and product by A). Casting out 9’s. B). Casting out 11’s: 6420 + 1819 + 7721 + 4328 + 3344 + 6749 = 30,380, and 728614 * 24637 = 1790586312. 3. Write half a page on the mathematical accomplishments of Archimedes.

12. Sep 25. Sketch 4, Angela Killian, Emily Purdy: Text, page 91, Problems 1, 4, 5. Due Sep 30.

12. Sep 28. Sketch 5, Emily Mortensen and David Garner: Text, page 99, problem 2, plus: Discuss how to help students understand negative numbers. Due Oct 2.

13. Sep 30. Sketch 6, Stephanie Kirkham, Robert McClain: Text, page 105: 1 a,b,c; 2 a,b, Due Oct 5.

14. Oct 2. Sketch 7, Scott Mancuso. Text: Do either: A: Page 111, 2 and 6, or B: page 112, 3. Due Oct 7.

15. Oct 5. Sketch 8, Peter Busath. Text, page 119, 1,2,3. Due Oct 9 or 12.

16. Oct 12, Sketch 9, Sera Shin. Text, page 125, 1, 3, Due Oct 16.

17. Oct 14, Sketch 10, Jackie Lang, Marissa Norris. Homework: 1. Show how to solve x^2 + 2bx = 20 geometrically. 2. Why do you think Al-Khowarismi kept his quadratic in the form ax^2 + bx = c? Due Oct 19.

18. Oct 16, Sketch 11, Dan Gwynn. Text, page 137, 2, 3. Due Oct 21.

19. Oct 19, Sketch 12, Lindsey Hall and Kristine Jaussi. Homework: 1. Prove the three ways to find Pythagorean Triples work. 2. Give three different proofs for the Pythagorean theorem. 3. On Page 146, do problems 4d and 6 (why or why not). 4. Give Euclids own proof of the Pythagorean theorem, and explain how his description differs from today. Due Oct 23.

20. Oct 21, Sketch 13, Nathan Steiger. Homework, page 153, 1 abc, 2 abc. Due Oct 26.

21. Oct 26 Sketch 15, Chris Rackley. Homework. Do Project #1, p. 168.  This does not have to be done with only cardboard and if you wish (instead of bringing them all in) you may take ONE picture of ALL of the solids together,spaced evenly apart so that they are easily visible.  Turn this printed picture in on a piece of full-size paper with your name, date, etc. so that it can be graded.  Otherwise, make sure your names are on the solids when you turn them in. Due Friday, Oct 30.

22. Oct 28. Sketch 16, Jeff Tree. Homework, page 175, 2b, 3. Due Mon, Nov 2

Take Home Quiz. 1. Use the four place table of logarithms from class to solve:

2. Use the definition of logarithm, to prove the identity

3. Check the two operations below by: a. casting out 9's, and b. casting out 11's:

1. 641 + 7142 + 240 = 8023. 2. 22312 * 973 = 21707596. Is anything wrong here?

Due in Class Wed, Nov 4.

23. Oct 30, Sketch 17, Shirley Escobar and Daniel Fenn, Homework, page 184: 1, 2, 3. Due Wed, Nov 4.

24. Nov 2, Sketch 18, Chase Thomas. Homework, page 192, 1. Due Fri Nov 6.

25. Nov 4, Sketch 19, Abbigail Dement. Homework: For either hyperbolic space or the sphere, give a written explanation for why or why not each of Euclid’s five postulates is satisfied. Due Mon, Nov 9

26. Nov 6, Sketch 20, Stephen Ockerman. Homework, page 205: 1, 2 abc, 3. Due Wed, Nov 11.

27. Nov 9, Sketch 21, Jeff Anderson. Homework, page 213: 1, 4. Due Fri, Nov 13.

28. Nov 11, Sketch 22, Mallory Manning. Homework, Page 221: #1 for 50, 100, and 200 digits only, #2 c for pi, the square root of pi, pi squared, pi halves, and the square roots of the first six prime numbers. Due Mon, Nov 16.

29. Nov 13, Sketch 23, Tiffany Lindsey, Cristy Pfeifer. Homework, page 229, 1, 3a. due Wed, Nov 18

30. Nov 16, Sketch 24, Michael Collins. Homework, page 235, do 2 problems out of numbers 1-4. Due Friday, Nov 20.

31. Nov 18, Lisa Foote, Jessica Hartvigen, Sketch 25, Homework: Create a picture or model that represents infinity. Does your picture explain infinity as a set according to Cantor’s ideas, or as a “manner of speaking” as Gauss thought? Due Mon, Nov 23.

32. Nov 20, Sketch 14, Emily Dyke. Homework, page 161: 1, 2, 5 Due Tues, Nov 24.

33. 0.

34. Handout on finding derivatives. Due Mon, Apr 6.